Optimal. Leaf size=69 \[ \frac{b^2}{5 a^3 \left (a+b x^5\right )}-\frac{3 b^2 \log \left (a+b x^5\right )}{5 a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{2 b}{5 a^3 x^5}-\frac{1}{10 a^2 x^{10}} \]
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Rubi [A] time = 0.0469586, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 44} \[ \frac{b^2}{5 a^3 \left (a+b x^5\right )}-\frac{3 b^2 \log \left (a+b x^5\right )}{5 a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{2 b}{5 a^3 x^5}-\frac{1}{10 a^2 x^{10}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^{11} \left (a+b x^5\right )^2} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^2} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^3}-\frac{2 b}{a^3 x^2}+\frac{3 b^2}{a^4 x}-\frac{b^3}{a^3 (a+b x)^2}-\frac{3 b^3}{a^4 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=-\frac{1}{10 a^2 x^{10}}+\frac{2 b}{5 a^3 x^5}+\frac{b^2}{5 a^3 \left (a+b x^5\right )}+\frac{3 b^2 \log (x)}{a^4}-\frac{3 b^2 \log \left (a+b x^5\right )}{5 a^4}\\ \end{align*}
Mathematica [A] time = 0.0498613, size = 57, normalized size = 0.83 \[ \frac{a \left (\frac{2 b^2}{a+b x^5}-\frac{a}{x^{10}}+\frac{4 b}{x^5}\right )-6 b^2 \log \left (a+b x^5\right )+30 b^2 \log (x)}{10 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 62, normalized size = 0.9 \begin{align*} -{\frac{1}{10\,{a}^{2}{x}^{10}}}+{\frac{2\,b}{5\,{a}^{3}{x}^{5}}}+{\frac{{b}^{2}}{5\,{a}^{3} \left ( b{x}^{5}+a \right ) }}+3\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{4}}}-{\frac{3\,{b}^{2}\ln \left ( b{x}^{5}+a \right ) }{5\,{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02189, size = 95, normalized size = 1.38 \begin{align*} \frac{6 \, b^{2} x^{10} + 3 \, a b x^{5} - a^{2}}{10 \,{\left (a^{3} b x^{15} + a^{4} x^{10}\right )}} - \frac{3 \, b^{2} \log \left (b x^{5} + a\right )}{5 \, a^{4}} + \frac{3 \, b^{2} \log \left (x^{5}\right )}{5 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99615, size = 194, normalized size = 2.81 \begin{align*} \frac{6 \, a b^{2} x^{10} + 3 \, a^{2} b x^{5} - a^{3} - 6 \,{\left (b^{3} x^{15} + a b^{2} x^{10}\right )} \log \left (b x^{5} + a\right ) + 30 \,{\left (b^{3} x^{15} + a b^{2} x^{10}\right )} \log \left (x\right )}{10 \,{\left (a^{4} b x^{15} + a^{5} x^{10}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 131.09, size = 68, normalized size = 0.99 \begin{align*} \frac{- a^{2} + 3 a b x^{5} + 6 b^{2} x^{10}}{10 a^{4} x^{10} + 10 a^{3} b x^{15}} + \frac{3 b^{2} \log{\left (x \right )}}{a^{4}} - \frac{3 b^{2} \log{\left (\frac{a}{b} + x^{5} \right )}}{5 a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19763, size = 115, normalized size = 1.67 \begin{align*} -\frac{3 \, b^{2} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{4}} + \frac{3 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{3 \, b^{3} x^{5} + 4 \, a b^{2}}{5 \,{\left (b x^{5} + a\right )} a^{4}} - \frac{9 \, b^{2} x^{10} - 4 \, a b x^{5} + a^{2}}{10 \, a^{4} x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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